What is the greatest three-digit number that leaves a remainder of 2 when divided by 6, 9 and 12?

Explanation

1. 01

   The number is 2 more than a multiple of $\operatorname{LCM}(6, 9, 12) = 36$.

2. 02

   $999 \div 36 = 27$ remainder 27, so the largest three-digit multiple is $36 \times 27 = 972$.

3. 03

   Number = $972 + 2 = 974$.